Self-similar stochastic processes have many applications in signal processing (image analysis, speech synthesis, road/Ethernet traffic modeling...). These signals of varying sparsity can be reconstructed efficiently using variational methods. Since fractional derivatives are whitening operators for these processes, we formulate a continous inverse problem with gTV regularization and generalized fractional derivatives as regularization operators. We then discretize this problem in the basis of periodic fractional B-splines, and propose an algorithm to solve this discretized problem in an exact way.